I used python to test this out and it quickly became unwieldy, although I managed to get some results.
Multiplying consecutive primes starting from 2 and adding the last consecutive prime.
Code:
6 + 5 = 11 is prime
30 + 7 = 37 is prime
210 + 11 = 221 13 * 17
2310 + 13 = 2323 23 * 101
30030 + 17 = 30047 is prime
510510 + 19 = 510529 is prime
9699690 + 23 = 9699713 is prime
223092870 + 29 = 223092899 127 * 1231 * 1427 False
6469693230 + 31 = 6469693261 107 * 787 * 76829 False
Multiplying consecutive primes and then adding any greater prime.
Code:
2x3 = 6
6 + 5 = 11 -> is prime
6 + 7 = 13 -> is prime
6 + 11 = 17 -> is prime
6 + 13 = 19 -> is prime
6 + 17 = 23 -> is prime
2x3x5 = 30
30 + 7 = 37 -> is prime
30 + 11 = 41 -> is prime
30 + 13 = 43 -> is prime
30 + 17 = 47 -> is prime
30 + 19 = 49 -> 7 * 7
2x3x5x7 = 210
210 + 11 = 221 -> 13 * 17
210 + 13 = 223 -> is prime
210 + 17 = 227 -> is prime
210 + 19 = 229 -> is prime
210 + 23 = 233 -> is prime
2x3x5x7x11 = 2310
2310 + 13 = 2323 -> 23 * 101
2310 + 17 = 2327 -> 13 * 179
2310 + 19 = 2329 -> 17 * 137
2310 + 23 = 2333 -> is prime
2310 + 29 = 2339 -> is prime
2x3x5x7x11x13 = 30030
30030 + 17 = 30047 -> is prime
30030 + 19 = 30049 -> 151 * 199
30030 + 23 = 30053 -> 41 * 733
30030 + 29 = 30059 -> is prime
30030 + 31 = 30061 -> 23 * 1307
2x3x5x7x11x13x17 = 510510
510510 + 19 = 510529 -> is prime
510510 + 23 = 510533 -> 83 * 6151
510510 + 29 = 510539 -> 31 * 43 * 383 False
510510 + 31 = 510541 -> 307 * 1663
510510 + 37 = 510547 -> 139 * 3673
2x3x5x7x11x13x17x19 = 9699690
9699690 + 23 = 9699713 -> is prime
9699690 + 29 = 9699719 -> 47 * 47 * 4391 False
9699690 + 31 = 9699721 -> 23 * 421727
9699690 + 37 = 9699727 -> is prime
9699690 + 41 = 9699731 -> is prime
2x3x5x7x11x13x17x19x23 = 223092870
223092870 + 29 = 223092899 -> 127 * 1231 * 1427 False
223092870 + 31 = 223092901 -> 43 * 5188207
223092870 + 37 = 223092907 -> is prime
223092870 + 41 = 223092911 -> 29 * 29 * 265271 False
223092870 + 43 = 223092913 -> 6971 * 32003
I thought I might as well post it.
Could only find 6 occurrences where this was False. This is a small sample, but I think you were on to something, at least for a quick way of finding a number which is likely prime or semi-prime.